generic grey value functions and the line of extremal slope joshua stough math 210, jim damon may 5,...
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![Page 1: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/1.jpg)
Generic Grey Value Functions and the Line of Extremal Slope
Joshua Stough
MATH 210, Jim Damon
May 5, 2003
![Page 2: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/2.jpg)
Motivation: determine properties of the edge line for generic grey value surfaces
G(x,y) = (x, y, g(x,y)); image graph
g(x,y) =
•Previous work on describing and detecting edge lines uses idealized/degenerate models of g.
•Mathematical approach: determine properties of graph of generic smooth g.
•Lens distortion, noise conceivably lead to generic g.
2
22
''*)','(4
1 4
])'()'[(
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t
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dydxeyxft
![Page 3: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/3.jpg)
Outline
•Definitions
•Generic properties of the edge line
•Generic properties of the evolving under linear diffusion
•The hypersurface of extremal slope, and selected results
![Page 4: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/4.jpg)
Canny edges
H(g), g = 0, g 0
![Page 5: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/5.jpg)
, P, g
![Page 6: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/6.jpg)
Generic properties of contains points only of negative Gaussian curvature
meets P tangentially at isolated points and and P are smooth at these points. The level curves at P are tranverse to both.
•The only singular points of are tansverse double points corresponding to Morse saddle points of G
g does not meet P and intersects transversely at isolated points
has isolated curvature extrema corresponding to A3 circles of curvature
![Page 7: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/7.jpg)
Outline
•Definitions
•Generic properties of the edge line
•Generic properties of the evolving under linear diffusion
•The hypersurface of extremal slope, and selected results
![Page 8: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/8.jpg)
Generic Evolutions of on families of diffused greyvalue surfaces
If f is analytic on a domain U, then a point z0 on the boundary U is called regular if f extends to be a analytic function on an open set containing U and also the point z0 (Krantz 1999, p. 119). Basically, z0 fits (is consistent with) its surroundings.
![Page 9: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/9.jpg)
Edge line evolution for coronal CT scanSigma = sqrt(2*t)
![Page 10: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/10.jpg)
Morse saddle stability implies stability g(x,y) = x^2 - y^2 + t*x*y^3 (H.O.T)
, w/o h.o.t, H(g), g = 8x2 – 8y2 = 0 x = y
•P, w/ h.o.t, det(H) = 0 x = (4 + 9*t2y4) / (12*ty)
•P not on pure Morse saddle
![Page 11: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/11.jpg)
Evolution of the Edge Line in Forming a rhamphoid cusp: gt = x^2 + 6*t*y + y^3 + 2*t
![Page 12: Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003](https://reader034.vdocuments.net/reader034/viewer/2022042821/56649d385503460f94a1258e/html5/thumbnails/12.jpg)
The hypersurface of extremal slope
is a hypersurface with isolated singular points.
•The generic geometry of ( \ {x : g = 0}) (punctured set) is the same as a general hypersurface ( without the closure?).
•At singular points of g of type Ak, has A3k-2 points ( has non-simple critical points at Dk4, E6,7,8 points of g.)
•Generically (codim 0) has only isolated A1 points at A1 points of g.
• In codim 1, can have A1 points at regular points of g, and can also have A4 points at A2 points of g.